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1.
The one-lie Rényi-Ulam liar game is a two-player perfect information zero-sum game, lasting q rounds, on the set [ n]?{1,…, n}. In each round Paul chooses a subset A⊆[ n] and Carole either assigns one lie to each element of A or to each element of [ n]? A. Paul wins the original (resp. pathological) game if after q rounds there is at most one (resp. at least one) element with one or fewer lies. We exhibit a simple, unified, optimal strategy for Paul to follow in both games, and use this to determine which player can win for all q, n and for both games. 相似文献
2.
We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x∈[ n], and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [ n]= A0∪?∪ At−1 and asks for a such that x∈ Aa. Carole responds with some b, and if a≠ b, then x accumulates a lie ( a, b). Carole's string of lies for x must be in the channel C. Paul wins if he determines x within q rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space [ n] for which Paul can guarantee finding the distinguished element is as q→∞, where Ek( C) is the number of lie strings in C of maximum length k. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel C. 相似文献
3.
The original Erd
s—Rényi theorem states that max 0kn∑ k+[clogn]i=k+1Xi/[ clog n]→α( c), c>0, almost surely for i.i.d. random variables { Xn, n1} with mean zero and finite moment generating function in a neighbourhood of zero. The latter condition is also necessary for the Erd
s—Rényi theorem, and the function α( c) uniquely determines the distribution function of X1. We prove that if the normalizing constant [ c log n] is replaced by the random variable ∑ k+[clogn]i=k+1( X2i+1), then a corresponding result remains true under assuming only the exist first moment, or that the underlying distribution is symmetric. 相似文献
4.
In this paper, we initiate the study of Ehrenfeucht–Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call the Ehrenfeucht–Fraïssé problem. Given nω as a parameter, and two relational structures and from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the n-round EF game ? We provide algorithms for solving the Ehrenfeucht–Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n. 相似文献
5.
For an [ n, k, d] q code
with k3, gcd( d, q)=1, the diversity of
is defined as the pair (Φ 0,Φ 1) with All the diversities for [ n, k, d] q codes with k3, d−2 (mod q) such that Ai=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated. 相似文献
6.
We discuss the existence of a diffeomorphism such that | where are closed differential forms and 2kn. Our main results (the case k=n having been handled by Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] and Dacorogna and Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) are that- – when n is even and k=2, under some natural non-degeneracy condition, we can prove the existence of such diffeomorphism satisfying Dirichlet data on the boundary of a bounded open set and the natural Hölder regularity; at the same time we get Darboux theorem with optimal regularity;
- – we are also able to handle the degenerate cases when k=2 (in particular when n is odd), k=n−1 and some cases where 3kn−2.
Résumé
Nous montrons l'existence d'un difféomorphisme
satisfaisant
où
sont des formes différentielles fermées et 2
kn. Nos résultats principaux (le cas
k=
n a été discuté notamment dans Moser [J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965) 286–294] et Dacorogna et Moser [B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990) 1–26]) sont les suivants.
- – Si n est pair, k=2 et sous des conditions naturelles de non dégénérescence, nous montrons l'existence et la régularité dans les espaces de Hölder d'un tel difféomorphisme satisfaisant de plus une condition de Dirichlet. On obtient aussi le théorème de Darboux avec la régularité optimale.
- – Par ailleurs quand k=2 et n est impair ou k=n−1, ainsi que quelques cas particuliers où 3kn−2, nous montrons l'existence locale d'un tel difféomorphisme satisfaisant, en outre, des conditions de Cauchy.
Keywords: Darboux theorem; Symplectic forms; Pullback; Hölder regularity
相似文献
7.
Es werden nichtlineare Differentialgleichungen der Form (∂
u/∂
t) +
Au =
f(
u) in
n × [0, ∞) betrachtet. Dabei ist
A ein positiv definiter elliptischer Differentialoperator 2
m-ter Ordnung und
f eine nichtlineare glatte Funktion, die nicht schneller als ¦
u ¦
q wächst, wobei
q < 1 + [4
m/(
n − 2
m)] für
n > 2
m und
q < ∞ für
n 2
m, und deren Ableitungen polynomials Wachstum besitzen. Auβerdem besitze
f eine nichtpositive Stammfunktion. Unter diesen Voraussetzungen wird durch Betrachtung der zugehörigen Integralgleichung in
Lv
n und unter Benutzung der Sobolevräume gebrochener Ordnung die globale klassische Lösbarkeit des Cauchyproblems für glatte Anfangswerte gezeigt.
相似文献
8.
The grid graph is the graph on [
k]
n
={0,...,
k–1}
n
in which
x=(
x
i
)
1
n
is joined to
y=(
y
i
)
1
n
if for some
i we have |
x
i
–y
i
|=1 and
x
j
=
y
j
for all
ji. In this paper we give a lower bound for the number of edges between a subset of [
k]
n
of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that if
A[
k]
n
satisfies
k
n
/4|
A|3
k
n
/4 then there are at least
k
n–1
edges between
A and its complement.Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.We also give a best possible upper bound for the number of edges spanned by a subset of [
k]
n
of given cardinality. In particular, for
r=1,...,
k we show that if
A[
k]
n
satisfies |
A|
r
n
then the subgraph of [
k]
n
induced by
A has average degree at most 2
n(1–1/
r).Research partially supported by NSF Grant DMS-8806097
相似文献
9.
Using Nuttall's compact formula for the [
n,
n − 1] Pad'e approximant, the authors show that there is a natural connection between the Padé approximants of a series of Stieltjes and orthogonal polynomials. In particular, we obtain the precise error formulas. The [
n,
n − 1] Padé approximant in this case is just a Gaussian quadrature of the Stieltjes integral. Hence, analysis of the error is now possible and under very mild conditions it is shown that the [
n,
n +
j],
j − 1, Padé approximants converge to the Stieltjes integral.
相似文献
10.
Le nombre maximal de lignes de matrices seront désignées par:
1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;
2. (b) S0(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y1, y2, y3, ou y1 = y2 ≠ y3;
3. (c) T0(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.
La fonction
T0(
k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que
T0(
k, λ) max(λ + 2, (
k − λ)
2 +
k − λ + 1). La fonction
S0(
k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant
T0(
k, λ),
S0(
k, λ), dans la remarque 9 on généralise le problème concernant
R(
k, λ). La fonction
R(
k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que
R(
k, λ)
S0(
k, λ)
T0(
k, λ) d'où découle en particulier:
R(
k, λ) λ + 2 pour λ
k + 1 − (
k + 2)
1/2;
R(
k, λ) = 0(
k2) pour
k − λ = 0(
k);
R(
k, λ) (
k − 1)
2 − (
k + 2) pour
k 1191.
相似文献